In this data structure and algorithms tutorial, you will learn everything you need to know about the Heap Data Structure
Heap data structure is a complete binary tree that satisfies the heap property, where any given node is
Max-heap
Min-heap
This type of data structure is also called a binary heap.
Some of the important operations performed on a heap are described below along with their algorithms.
Heapify is the process of creating a heap data structure from a binary tree. It is used to create a Min-Heap or a Max-Heap.
Initial Array
Create a complete binary tree from the array
Complete binary tree
Start from the first index of non-leaf node whose index is given by n/2 - 1
.
Start from the first on leaf node
i
as largest
.2i + 1
and the right child is given by 2i + 2
.leftChild
is greater than currentElement
(i.e. element at ith
index), set leftChildIndex
as largest.rightChild
is greater than element in largest
, set rightChildIndex
as largest
.Swap largest
with currentElement
Swap if necessary
Algorithm
Heapify(array, size, i)
set i as largest
leftChild = 2i + 1
rightChild = 2i + 2
if leftChild > array[largest]
set leftChildIndex as largest
if rightChild > array[largest]
set rightChildIndex as largest
swap array[i] and array[largest]
To create a Max-Heap:
MaxHeap(array, size)
loop from the first index of non-leaf node down to zero
call heapify
For Min-Heap, both leftChild
and rightChild
must be larger than the parent for all nodes.
Algorithm for insertion in Max Heap
If there is no node,
create a newNode.
else (a node is already present)
insert the newNode at the end (last node from left to right.)
heapify the array
Insert the new element at the end of the tree.
Insert at the end
Heapify the tree.
Heapify the array
For Min Heap, the above algorithm is modified so that parentNode
is always smaller than newNode
.
Algorithm for deletion in Max Heap
If nodeToBeDeleted is the leafNode
remove the node
Else swap nodeToBeDeleted with the lastLeafNode
remove noteToBeDeleted
heapify the array
Select the element to be deleted.Select the element to be deleted
Swap it with the last element.
Swap with the last element
Remove the last element.
Remove the last element
Heapify the tree.
Heapify the array
For Min Heap, above algorithm is modified so that both childNodes
are greater smaller than currentNode
.
Peek operation returns the maximum element from Max Heap or minimum element from Min Heap without deleting the node.
For both Max heap and Min Heap
return rootNode
Extract-Max returns the node with maximum value after removing it from a Max Heap whereas Extract-Min returns the node with minimum after removing it from Min Heap.
Python
# Max-Heap data structure in Python
def heapify(arr, n, i):
largest = i
l = 2 * i + 1
r = 2 * i + 2
if l < n and arr[i] < arr[l]:
largest = l
if r < n and arr[largest] < arr[r]:
largest = r
if largest != i:
arr[i],arr[largest] = arr[largest],arr[i]
heapify(arr, n, largest)
def insert(array, newNum):
size = len(array)
if size == 0:
array.append(newNum)
else:
array.append(newNum);
for i in range((size//2)-1, -1, -1):
heapify(array, size, i)
def deleteNode(array, num):
size = len(array)
i = 0
for i in range(0, size):
if num == array[i]:
break
array[i], array[size-1] = array[size-1], array[i]
array.remove(num)
for i in range((len(array)//2)-1, -1, -1):
heapify(array, len(array), i)
arr = []
insert(arr, 3)
insert(arr, 4)
insert(arr, 9)
insert(arr, 5)
insert(arr, 2)
print ("Max-Heap array: " + str(arr))
deleteNode(arr, 4)
print("After deleting an element: " + str(arr))
Java
// Max-Heap data structure in Java
import java.util.ArrayList;
class Heap {
void heapify(ArrayList<Integer> hT, int i) {
int size = hT.size();
int largest = i;
int l = 2 * i + 1;
int r = 2 * i + 2;
if (l < size && hT.get(l) > hT.get(largest))
largest = l;
if (r < size && hT.get(r) > hT.get(largest))
largest = r;
if (largest != i) {
int temp = hT.get(largest);
hT.set(largest, hT.get(i));
hT.set(i, temp);
heapify(hT, largest);
}
}
void insert(ArrayList<Integer> hT, int newNum) {
int size = hT.size();
if (size == 0) {
hT.add(newNum);
} else {
hT.add(newNum);
for (int i = size / 2 - 1; i >= 0; i--) {
heapify(hT, i);
}
}
}
void deleteNode(ArrayList<Integer> hT, int num)
{
int size = hT.size();
int i;
for (i = 0; i < size; i++)
{
if (num == hT.get(i))
break;
}
int temp = hT.get(i);
hT.set(i, hT.get(size-1));
hT.set(size-1, temp);
hT.remove(size-1);
for (int j = size / 2 - 1; j >= 0; j--)
{
heapify(hT, j);
}
}
void printArray(ArrayList<Integer> array, int size) {
for (Integer i : array) {
System.out.print(i + " ");
}
System.out.println();
}
public static void main(String args[]) {
ArrayList<Integer> array = new ArrayList<Integer>();
int size = array.size();
Heap h = new Heap();
h.insert(array, 3);
h.insert(array, 4);
h.insert(array, 9);
h.insert(array, 5);
h.insert(array, 2);
System.out.println("Max-Heap array: ");
h.printArray(array, size);
h.deleteNode(array, 4);
System.out.println("After deleting an element: ");
h.printArray(array, size);
}
}
C programming
// Max-Heap data structure in C
#include <stdio.h>
int size = 0;
void swap(int *a, int *b)
{
int temp = *b;
*b = *a;
*a = temp;
}
void heapify(int array[], int size, int i)
{
if (size == 1)
{
printf("Single element in the heap");
}
else
{
int largest = i;
int l = 2 * i + 1;
int r = 2 * i + 2;
if (l < size && array[l] > array[largest])
largest = l;
if (r < size && array[r] > array[largest])
largest = r;
if (largest != i)
{
swap(&array[i], &array[largest]);
heapify(array, size, largest);
}
}
}
void insert(int array[], int newNum)
{
if (size == 0)
{
array[0] = newNum;
size += 1;
}
else
{
array[size] = newNum;
size += 1;
for (int i = size / 2 - 1; i >= 0; i--)
{
heapify(array, size, i);
}
}
}
void deleteRoot(int array[], int num)
{
int i;
for (i = 0; i < size; i++)
{
if (num == array[i])
break;
}
swap(&array[i], &array[size - 1]);
size -= 1;
for (int i = size / 2 - 1; i >= 0; i--)
{
heapify(array, size, i);
}
}
void printArray(int array[], int size)
{
for (int i = 0; i < size; ++i)
printf("%d ", array[i]);
printf("\n");
}
int main()
{
int array[10];
insert(array, 3);
insert(array, 4);
insert(array, 9);
insert(array, 5);
insert(array, 2);
printf("Max-Heap array: ");
printArray(array, size);
deleteRoot(array, 4);
printf("After deleting an element: ");
printArray(array, size);
}
C++
// Max-Heap data structure in C++
#include <iostream>
#include <vector>
using namespace std;
void swap(int *a, int *b)
{
int temp = *b;
*b = *a;
*a = temp;
}
void heapify(vector<int> &hT, int i)
{
int size = hT.size();
int largest = i;
int l = 2 * i + 1;
int r = 2 * i + 2;
if (l < size && hT[l] > hT[largest])
largest = l;
if (r < size && hT[r] > hT[largest])
largest = r;
if (largest != i)
{
swap(&hT[i], &hT[largest]);
heapify(hT, largest);
}
}
void insert(vector<int> &hT, int newNum)
{
int size = hT.size();
if (size == 0)
{
hT.push_back(newNum);
}
else
{
hT.push_back(newNum);
for (int i = size / 2 - 1; i >= 0; i--)
{
heapify(hT, i);
}
}
}
void deleteNode(vector<int> &hT, int num)
{
int size = hT.size();
int i;
for (i = 0; i < size; i++)
{
if (num == hT[i])
break;
}
swap(&hT[i], &hT[size - 1]);
hT.pop_back();
for (int i = size / 2 - 1; i >= 0; i--)
{
heapify(hT, i);
}
}
void printArray(vector<int> &hT)
{
for (int i = 0; i < hT.size(); ++i)
cout << hT[i] << " ";
cout << "\n";
}
int main()
{
vector<int> heapTree;
insert(heapTree, 3);
insert(heapTree, 4);
insert(heapTree, 9);
insert(heapTree, 5);
insert(heapTree, 2);
cout << "Max-Heap array: ";
printArray(heapTree);
deleteNode(heapTree, 4);
cout << "After deleting an element: ";
printArray(heapTree);
}
#datastructures #algorithms